Abstract

Analytical or numerical solutions are given for the kinetic equations describing the time behavior of a solid photochromic film for the cases of photocoloration, photobleaching, and thermal bleaching. A procedure to evaluate relative quantum yields and absorption cross sections from the comparison of experimental results with theoretical solutions is demonstrated by an example using 6-nitro-8 piperidinomethyl-1′,3′,3′-trimethylspiro-[1H-benzopyran-2,2′-indoline] as a photochromic compound. The theoretical results are presented in a suitably normalized form to be applicable to a broad range of experimental situations. The agreement between calculated and measured results is good as long as strictly first-order reaction mechanisms are obeyed.

© 1973 Optical Society of America

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References

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  1. G. G. Dorion, A. F. Wiebe, Photochromism, Optical and Photographic Applications (The Focal Press, London, 1970).
  2. C. T. Slack, Optica Acta 17, 547 (1970).
    [CrossRef]
  3. W. J. Tomlinson, Appl. Opt. 11, 823 (1972).
    [CrossRef] [PubMed]
  4. R. C. Bertelson, in Photochromism, G. H. Brown, Ed. (Wiley, New York, 1971), p. 190.
  5. I. P. Kaminow, L. W. Stulz, E. A. Chandross, C. A. Pryde, Appl. Opt. 11, 1563 (1972).
    [CrossRef] [PubMed]
  6. Registered trademark, Rohm & Haas Co.
  7. Registered trademark, Bexford Ltd.
  8. L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
    [CrossRef]

1972 (2)

1970 (1)

C. T. Slack, Optica Acta 17, 547 (1970).
[CrossRef]

1967 (1)

L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
[CrossRef]

Bertelson, R. C.

R. C. Bertelson, in Photochromism, G. H. Brown, Ed. (Wiley, New York, 1971), p. 190.

Chandross, E. A.

Davis, R. B.

L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
[CrossRef]

Dorion, G. G.

G. G. Dorion, A. F. Wiebe, Photochromism, Optical and Photographic Applications (The Focal Press, London, 1970).

Kaminow, I. P.

Nicholson, J.

L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
[CrossRef]

Pryde, C. A.

Slack, C. T.

C. T. Slack, Optica Acta 17, 547 (1970).
[CrossRef]

Stulz, L. W.

Taylor, L. D.

L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
[CrossRef]

Tomlinson, W. J.

Wiebe, A. F.

G. G. Dorion, A. F. Wiebe, Photochromism, Optical and Photographic Applications (The Focal Press, London, 1970).

Appl. Opt. (2)

Optica Acta (1)

C. T. Slack, Optica Acta 17, 547 (1970).
[CrossRef]

Tetrahedron Lett. (1)

L. D. Taylor, J. Nicholson, R. B. Davis, Tetrahedron Lett. 17, 1585 (1967).
[CrossRef]

Other (4)

Registered trademark, Rohm & Haas Co.

Registered trademark, Bexford Ltd.

G. G. Dorion, A. F. Wiebe, Photochromism, Optical and Photographic Applications (The Focal Press, London, 1970).

R. C. Bertelson, in Photochromism, G. H. Brown, Ed. (Wiley, New York, 1971), p. 190.

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Figures (8)

Fig. 1
Fig. 1

Optical density vs time in normalized units for a photo-chromic film irradiated at wavelength λ1 with equal absorption cross sections of both colorless and colored state (σA1 = σB1). The parameter is [k + r exp(−qx)]/(k + r), which is approximately equal to the initial film transmittance at λ1.

Fig. 2
Fig. 2

Optical density vs time in normalized units for the general case σA1σB1. Initial film transmittance y = 0.1. For the definition of parameters r and K, see Sec. II C.

Fig. 3
Fig. 3

Optical density vs time in normalized units for the general case σA1σB1. Initial film transmittance y = 0.2.

Figure 4
Figure 4

Optical density vs time in normalized units for the general case σA1σB1. Initial film transmittance at λ1y = 0.3. The dashed line is an experimental coloration curve for a 13.5-μ thick photochromic film, optimally fitted to the calculated curves by shifting along the time axis.

Fig. 5
Fig. 5

Optical density vs time in normalized units for the general case σA1σB1. Initial film transmittance y = 0.4.

Fig. 6
Fig. 6

Absorption spectra of a photochromic film before (solid line, film thickness 14 μ) and after (dashed line, film thickness 9 μ) irradiation with 0.85-mW/cm2 uv light at λ1 = 367 nm for 240 sec. The insert shows the photochromic spiropyran compound used for the experiments.

Fig. 7
Fig. 7

Photocoloration curve of a 13.5-μ thick film (density vs time), irradiated at 367 nm with 0.85 mW/cm2. The same curve is replotted in Fig. 4 in normalized form (dashed line).

Fig. 8
Fig. 8

Thermal and optical bleaching curves for photochromic films of 14-μ and 13-μ thickness, respectively. Bleaching radiation at 580 nm, 1.05 mW/cm2.

Equations (31)

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N B t = σ A 1 ϕ c N A I 1 - σ B 1 ϕ b 1 N B I 1 - σ B 2 ϕ b 2 N B I 2 - k N B ,
I 1 / x = - σ A 1 N A I 1 - σ B 1 N B I 1 - α I 1
I 2 / x = - σ B 2 N B I 2 .
N B / t = - k N B .
D ( t ) = D 0 exp ( - k t ) ,
D 0 = M σ B 2 0 L N B 0 ( x ) d x .
N B / t = - σ B 2 ϕ b 2 N B I 2 - k N B ,
I 2 / x = - σ B 2 N B I 2 .
I 2 = I 20 exp [ - σ B 2 0 x N B ( x , t ) d x ] = I 20 exp ( - D / M ) .
2 D x t = [ - σ B 2 ϕ b 2 I 20 exp ( - D / M ) - k ] ( D / x ) .
D t = M σ B 2 ϕ b 2 I 20 [ exp ( - D / M ) - 1 ] - k D .
D ( x , t ) = log { 1 + ( 10 D 0 - 1 ) exp [ - ( σ B 2 ϕ b 2 I 20 + k ) t ] } .
N B / t = σ A 1 ϕ c N A I 1 - σ B 1 ϕ b 1 N B I 1 - k N B ,
I 1 / x = - σ A 1 N A I 1 - σ B 1 N B I 1 - α I 1 .
D ( x , t ) = M W ( E 1 [ ( k + r ) t ] - E 1 { [ k + r exp ( - q x ) ] t } - ln k + r exp ( - q x ) k + r )
E 1 ( x ) = 1 exp ( - x t ) t d t , r = ( σ A 1 ϕ c + σ B 1 ϕ b 1 ) I 10 , q = σ A 1 N T + α , W = ( σ B 2 N T / q ) ϕ c / ( ϕ c + ϕ b 1 ) .
I 1 = I 10 exp [ ( - σ A 1 N T + α ) x ] exp ( σ A 1 - σ B 1 σ B 2 D / M ) .
u ( y , τ ) = exp [ ( σ A 1 - σ B 1 ) 0 x N B ( x , t ) d x ] = exp { [ ( σ A 1 - σ B 1 ) / σ B 2 ] D / M } y = exp [ - ( σ A 1 N T + α ) x ] = exp ( - q x ) τ = r t .
2 u y τ = 1 u u τ u y - k r u y + u ( K u - y u y ) ,
K = [ σ A 1 ϕ c ( σ B 1 - σ A 1 ) N T ] / [ ( σ A 1 ϕ c + σ B 1 ϕ b 1 ) q ] .
( k / r ) ( d u / d y ) = u [ K u - y ( d u / d y ) ] .
ln v y + K K + 1 ln k + r ( K + 1 ) k + r ( K + 1 ) v = 0
D sat = K σ B 2 σ B 1 - σ A 1 log k + r ( K + 1 ) k · 10 [ ( σ B 1 - σ A 1 ) / σ B 2 ] D sat + r ( K + 1 ) y ,
N B sat = q σ B 1 - σ A 1 K r y k · 10 ( σ B 1 - σ A 1 ) D sat / σ B 2 + r y .
q = σ A 1 N T + α σ A 1 N T = 8.36 × 10 2 cm - 1 , σ A 1 = 152 × 10 - 17 cm 2 , ( σ B 1 - σ A 1 ) / σ B 2 = 0.644.
k = 4.74 × 10 - 4 sec - 1 ( t < 700 sec ) σ B 2 ϕ b 2 I 20 = 1.04 × 10 - 2 sec - 1 . ( t < 150 sec )
r = 9.3 × 10 - 3 sec - 1 , k / r = 0.051.
k = 4.74 × 10 - 4 sec - 1 , σ A 1 = 1.52 × 10 - 17 cm 2 , σ B 1 = ( 1.52 + 0.503 / ϕ c ) × 10 - 17 cm 2 , σ B 2 = ( 0.781 / ϕ c ) × 10 - 17 cm 2 , ϕ b 1 = ϕ c ( 0.389 - ϕ c ) / ( 0.331 + ϕ c ) , ϕ b 2 = 0.437 ϕ c .
D sat = 0.85 0.644 log × 4.74 × 10 - 4 + 9.3 × 10 - 3 × 1.85 4.74 × 10 - 4 × 10 0.644 D sat + 9.3 × 10 - 3 × 1.85 × 0.32 .
D sat = 0.57
N B sat = 1.01 × 10 20 × ϕ c cm - 3 = 1.83 ϕ c N T .

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